, where \(\mathbf {a}(t)\)  is the time dependent axis of rotation and \(\mathbf {L}\)  the vector of the infinitesimal generators of the Lie group of rotations [1]}. In this case, the exponential is a matrix exponential; vectorpath works in the approximation that \(e^Xe^Y = e^Z = e^{X+Y}\)  for matrices even if X and Y don't commute. This is not true in general, but can be assumed true in first approximation for infinitesimally small X and Y, as one can see from the Baker-Campbell-Hausdorff formula [1]}, in the case in which both X and Y are of order \(\epsilon \) :
\(Z &= X + Y + \frac{1}{2}[X, Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + ... \nonumber \\&= X + Y + \mathcal {O}(\epsilon ^2)\) 
